Hydrocyclone Application for Starch Concentration

Reference ID: MET-374E | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The hydrocyclone is a centrifugal liquid‑solid separator widely employed in the starch industry to concentrate starch slurries. By introducing the feed tangentially, a high‑speed vortex generates a radial acceleration that forces denser starch particles toward the underflow, while the lighter water‑rich phase exits as overflow. Accurate prediction of the cut size, separation efficiency, and pressure drop is essential for sizing cyclones, arranging them in batteries, and ensuring energy‑efficient operation.

Methodology & Formulas

The calculation follows the physics embedded in the reference Python code, expressed with standard engineering symbols.

1. Unit conversions

\[ Q_{s}= \frac{Q_{h}}{3600}\qquad\text{(volumetric flow rate in m³·s⁻¹)} \] \[ \mu = \mu_{cP}\cdot10^{-3}\qquad\text{(dynamic viscosity in Pa·s)} \] \[ \Delta\rho = \rho_{starch}-\rho_{water}\qquad\text{(density difference, kg·m⁻³)} \] \[ \theta = \frac{\pi}{180}\, \theta_{deg}\qquad\text{(cone half‑angle in radians)} \] \[ \sin\theta = \sin\!\left(\theta\right) \]

2. Hydraulic geometry

\[ A_{in}= \pi\left(\frac{d_{i}}{2}\right)^{2}\qquad\text{(inlet cross‑sectional area, m²)} \] \[ v_{in}= \frac{Q_{s}}{A_{in}}\qquad\text{(inlet velocity, m·s⁻¹)} \]

3. Cut size (d₅₀)

\[ d_{50}= K_{d50}\,\sqrt{\frac{\mu\,D_{c}}{\Delta\rho\,v_{in}\,\sin\theta}}\qquad\text{(median cut size, m)} \] where \(K_{d50}\) is a dimensionless empirical factor (≈ 0.55 for standard geometries).

4. Separation efficiency

\[ \eta = 1-\exp\!\left[-\left(\frac{d_{p}}{d_{50}}\right)^{N_{exp}}\right]\qquad\text{(fractional efficiency)} \] with \(d_{p}\) the particle diameter of interest and \(N_{exp}\) the exponential shape parameter (≈ 1.5 for starch slurries).

5. Pressure drop

\[ \Delta P_{Pa}= \tfrac{1}{2}\,\rho_{slurry}\,v_{in}^{2}\,\bigl(1+K_{loss}\bigr) \] \[ \Delta P_{bar}= \frac{\Delta P_{Pa}}{10^{5}}\qquad\text{(pressure drop, bar)} \] \(K_{loss}\) is the overall loss coefficient (≈ 7.5) and \(\rho_{slurry}\) the bulk slurry density.

Empirical Validity Checks

Parameter	Condition	Typical Range
Inlet‑to‑body diameter ratio	\(0.15 \le \dfrac{d_{i}}{D_{c}} \le 0.25\)	0.15 – 0.25
Volumetric flow rate	\(5 \le Q_{h} \le 20\) m³·h⁻¹	5 – 20 m³/h
Inlet velocity	\(3 \le v_{in} \le 10\) m·s⁻¹	3 – 10 m/s
Density difference	\(\Delta\rho > 50\) kg·m⁻³	> 50 kg/m³
Pressure drop	\(0.5 \le \Delta P_{bar} \le 8\) bar	0.5 – 8 bar

To optimize starch recovery and concentration, process engineers must monitor and control the following variables:

Feed solids concentration: Maintaining a consistent feed density is critical to prevent underflow plugging.
Pressure drop: Higher pressure drops generally improve separation efficiency but increase energy consumption and shear forces.
Vortex finder diameter: Adjusting this dimension directly influences the cut point and the split ratio between overflow and underflow.
Feed temperature: Viscosity changes in the starch slurry can significantly alter the separation performance.

Roping occurs when the underflow solids concentration exceeds the discharge capacity of the apex. To prevent this, consider these strategies:

Regularly inspect apex nozzles for wear, as enlarged openings can lead to excessive water loss and unstable flow patterns.
Implement automated feed density control loops to ensure the slurry does not exceed the design capacity of the cyclone bank.
Monitor the pressure differential across the cluster to detect early signs of flow restriction.

Because starch slurries can be abrasive depending on the presence of impurities, a proactive maintenance approach is required:

Perform visual inspections of the vortex finder and apex every 500 operating hours.
Measure the internal diameter of the apex nozzle monthly to track erosion rates.
Replace ceramic or polyurethane liners immediately if surface pitting or channeling is observed to prevent catastrophic failure of the cyclone housing.

Worked Example: Hydrocyclone Sizing for Corn Starch Clarification

A process engineer needs to evaluate the performance of a single hydrocyclone unit for concentrating a corn starch slurry. The goal is to estimate the cut size, separation efficiency for the target starch particles, and the required operating pressure.

Known Parameters:

Hydrocyclone diameter, \( D_{c} = 0.100 \ \text{m} \)
Inlet diameter, \( D_{in} = 0.020 \ \text{m} \)
Volumetric feed flow rate, \( Q_{h} = 10.000 \ \text{m}^{3}/\text{h} \)
Target particle size, \( d_{p} = 20.000 \ \mu\text{m} \)
Slurry dynamic viscosity, \( \mu_{cP} = 1.000 \ \text{cP} \)
Slurry bulk density, \( \rho_{slurry} = 1020.000 \ \text{kg}/\text{m}^{3} \)
Starch density, \( \rho_{starch} = 1500.000 \ \text{kg}/\text{m}^{3} \)
Water density, \( \rho_{water} = 1000.000 \ \text{kg}/\text{m}^{3} \)
Cone half-angle, \( \theta_{deg} = 20.000^{\circ} \)
Empirical constant for cut size, \( K_{d50} = 0.550 \)
Overall pressure loss coefficient, \( K_{loss} = 7.500 \)
Exponential shape parameter, \( N_{exp} = 1.500 \)

Step-by-Step Calculation:

Perform unit conversions and compute derived hydraulic parameters.
The flow rate is converted to SI units: \( Q_{s} = \frac{10.000}{3600} = 0.0027778 \ \text{m}^{3}/\text{s} \).
The dynamic viscosity: \( \mu = 1.000 \times 10^{-3} = 0.001000 \ \text{Pa}\cdot\text{s} \).
The density difference is: \( \Delta\rho = \rho_{starch} - \rho_{water} = 500.000 \ \text{kg}/\text{m}^{3} \).
The cone half-angle in radians: \( \theta = \frac{\pi}{180} \times 20.000 = 0.349066 \ \text{rad} \).
The inlet cross-sectional area is: \( A_{in} = \pi \left( \frac{0.020}{2} \right)^{2} = \pi \times (0.010)^{2} = 0.00031416 \ \text{m}^{2} \).
The inlet velocity is therefore: \( v_{in} = \frac{Q_{s}}{A_{in}} = \frac{0.0027778}{0.00031416} = 8.842 \ \text{m}/\text{s} \).
The sine of the cone angle is: \( \sin\theta = \sin(20.000^{\circ}) = 0.3420 \).
Calculate the cut size \( d_{50} \).
Using the provided correlation: \[ d_{50} = K_{d50} \cdot \sqrt{ \frac{\mu \cdot D_{c}}{\Delta\rho \cdot v_{in} \cdot \sin\theta} } \] \[ d_{50} = 0.550 \times \sqrt{ \frac{0.001000 \times 0.100}{500.000 \times 8.842 \times 0.3420} } = 0.550 \times \sqrt{6.614 \times 10^{-8}} = 0.550 \times 0.0002571 = 0.0001414 \ \text{m} \] The calculation yields a cut size of \( d_{50} = 141.4 \ \mu\text{m} \).
Estimate the separation efficiency for the target 20 μm particles.
Applying the exponential efficiency model with exponent \( N_{exp} = 1.5 \): \[ \eta = 1 - \exp\!\left[-\left( \frac{d_{p}}{d_{50}} \right)^{N_{exp}} \right] = 1 - \exp\!\left[-\left( \frac{20.000}{141.4} \right)^{1.5} \right] \] \[ \left( \frac{20.000}{141.4} \right)^{1.5} = (0.1414)^{1.5} = 0.0532, \quad \eta = 1 - \exp(-0.0532) = 0.0518 \] The result is a separation efficiency of \( \eta = 0.052 \) or \( 5.2\% \).
Compute the system pressure drop.
The pressure drop is calculated from the inlet dynamic head and loss coefficient: \[ \Delta P_{Pa} = \frac{1}{2} \cdot \rho_{slurry} \cdot v_{in}^{2} \cdot (1 + K_{loss}) \] \[ \Delta P_{Pa} = 0.5 \times 1020.000 \times (8.842)^{2} \times (1 + 7.500) = 510 \times 78.18 \times 8.5 = 338,900 \ \text{Pa} \] \[ \Delta P_{bar} = \frac{338,900}{10^{5}} = 3.389 \ \text{bar} \] This gives a pressure drop of \( \Delta P = 3.389 \ \text{bar} \).
Perform empirical validity checks.
The inlet diameter ratio \( D_{in}/D_{c} = 0.020/0.100 = 0.200 \) is within the typical range of 0.15–0.25.
The flow rate of \( 10.000 \ \text{m}^{3}/\text{h} \) is within the typical range for a 0.1 m hydrocyclone (5–20 m³/h).
The inlet velocity of \( 8.842 \ \text{m}/\text{s} \) is within the recommended range (3–10 m/s).
The density difference of \( 500.000 \ \text{kg}/\text{m}^{3} \) is sufficient for separation.
The calculated pressure drop of \( 3.389 \ \text{bar} \) is within an operable range.

Final Answer:

For the given hydrocyclone geometry and operating conditions:

The calculated cut size (\( d_{50} \)) is 141.4 μm.
The separation efficiency for 20 μm starch particles is 5.2%.
The required system pressure drop is 3.39 bar.

The low efficiency for the target particle size indicates that a single unit of this design is not effective. A battery of hydrocyclones in series or a design with a smaller cut size would be required for effective starch concentration.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle